Damping Optimization of Linear Vibrational Systems with a Singular Mass Matrix

We present two novel results for small damped oscillations described by the vector differential equation M x ¨ + C x ˙ + K x = 0 , where the mass matrix M can be singular, but standard deflation techniques cannot be applied. The first result is a novel formula for the solution X of the Lyapunov equation A T X + X A = − I , where A = A ( v ) is obtained from M , C ( v ) ∈ R n × n , and K ∈ R n × n , which are the so-called mass, damping, and stiffness matrices, respectively, and rank ( M ) = n − 1 . Here, C ( v ) is positive semidefinite with rank ( C ( v ) ) = 1 . Using the obtained formula, we propose a very efficient way to compute the optimal damping matrix. The second result was obtained for a different structure, where we assume that dim ( N ( M ) ) ≥ 1 and internal damping exists (usually a small percentage of the critical damping). For this structure, we introduce a novel linearization, i.e., a novel construction of the matrix A in the Lyapunov equation A T X + X A = − I , and a novel optimization process. The proposed optimization process computes the optimal damping C ( v ) that minimizes a function v ↦ trace ( Z X ) (where Z is a chosen symmetric positive semidefinite matrix) using the approximation function g ( v ) = c v + a v + b v , for the trace function f ( v ) ≐ trace ( Z X ( v ) ) . Both results are illustrated with several corresponding numerical examples.